Very unfortunate. If you want to sell it... let me know.
Boot sector is supposed to be on the hard drive... If the cable is bad, Windows cannot find it. If the hard drive is bad, you cannot find it.
But you might try formatting it in another drive...
Is this a SATA hard drive or an EIDE hard drive?
Is the hard drive that is installed the one that was clicking? Or was it the optical drive that was clicking.
Do you have another hard drive to test?
Does the Dell Windows disk start up in the optical drive?
It is possible that the new hard drive was damaged in shipment, or defective?
The next step is to try that drive in another computer... if you have a friend with patience...
Try it in a friend's laptop and format it with the friend's Windows software... then either install Windows there, or put it back in your computer and re-format it, and install windows if it will.
There are few hardware defects it could be... memory, hard drive, motherboard, CMOS batter, or cable...
We have to assume the hard drive is bad until you try it elsewhere...
You could try installing it in a USB enclosure, then formatting it on anther computer using any Windows setup... if it formats, then it should be detected in your Inspiron 5160. If the computer still does not detect the hard drive after it has been formatted in another machine, then you have another potentially serious issue... It all depends upon whether that drive is detected in another laptop, or can be formatted elsewhere.